CAIE Further Paper 2 2024 November — Question 1 4 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2024
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
Topic3x3 Matrices
TypeParameter values for unique solution
DifficultyStandard +0.8 This requires computing a 3×3 determinant with parameter k, setting it non-zero for unique solution, and providing geometric interpretation (three planes meeting at a point). While systematic, it combines matrix theory, algebraic manipulation, and geometric understanding beyond routine A-level, typical of Further Maths content.
Spec4.03s Consistent/inconsistent: systems of equations
1 Find the set of values of \(k\) for which the system of equations $$\begin{array} { r } x + 5 y + 6 z = 1 \\ k x + 2 y + 2 z = 2 \\ - 3 x + 4 y + 8 z = 3 \end{array}$$ has a unique solution and interpret this situation geometrically.
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(\begin{vmatrix} 1 & 5 & 6 \\ k & 2 & 2 \\ -3 & 4 & 8 \end{vmatrix} = \begin{vmatrix} 2 & 2 \\ 4 & 8 \end{vmatrix} - 5\begin{vmatrix} k & 2 \\ -3 & 8 \end{vmatrix} + 6\begin{vmatrix} k & 2 \\ -3 & 4 \end{vmatrix} = 8 - 5(8k+6) + 6(4k+6)\)M1A1 Evaluates determinant
\(k \neq \frac{7}{8}\)A1
Three planes intersect at a *unique* point.B1
Total: 4 
## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{vmatrix} 1 & 5 & 6 \\ k & 2 & 2 \\ -3 & 4 & 8 \end{vmatrix} = \begin{vmatrix} 2 & 2 \\ 4 & 8 \end{vmatrix} - 5\begin{vmatrix} k & 2 \\ -3 & 8 \end{vmatrix} + 6\begin{vmatrix} k & 2 \\ -3 & 4 \end{vmatrix} = 8 - 5(8k+6) + 6(4k+6)$ | M1A1 | Evaluates determinant |
| $k \neq \frac{7}{8}$ | A1 | |
| Three planes intersect at a *unique* point. | B1 | |
| **Total: 4** | | |

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1 Find the set of values of $k$ for which the system of equations

$$\begin{array} { r } 
x + 5 y + 6 z = 1 \\
k x + 2 y + 2 z = 2 \\
- 3 x + 4 y + 8 z = 3
\end{array}$$

has a unique solution and interpret this situation geometrically.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2024 Q1 [4]}}
This paper (8 questions)
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Q1 4 Q2 6 Q3 12 Q4 9 Q5 10 Q6 10 Q7 10 Q8 14